Final answer:
The machinist must manufacture the disk with a radius between 17.799 cm and 17.882 cm to stay within the error tolerance of ± 5 cm², meaning the radius needs to be controlled very closely, within approximately 0.042 cm of the ideal radius.
Step-by-step explanation:
The machinist needs to control the radius to make a circular metal disk with an area of 1000 cm2. The formula for the area of a circle is A = πr². The ideal radius can be found by rearranging the formula to find r: r = √(A/π).
Given the area is 1000 cm2, the ideal radius would be r = √(1000 cm2/π) which approximately equals 17.841 cm. Due to the error tolerance of ± 5 cm2, we need to find the minimum and maximum radius the machinist can use without exceeding the tolerance. Using the formula again:
- For the minimum area (995 cm2), rmin = √(995 cm2/π) ≈ 17.799 cm.
- For the maximum area (1005 cm2), rmax = √(1005 cm2/π) ≈ 17.882 cm.
The machinist must control the radius within these bounds, rounding to the nearest hundred thousandth, which gives us 17.799 cm to 17.882 cm. The difference between the ideal and the minimum or maximum is approximately 0.042 cm, showing how close the machinist must control the radius to stay within the tolerance.